Reliable Methods for Computer Simulation: Error Control and Posteriori Estimates

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Elsevier, Sep 4, 2004 - Mathematics - 316 pages
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Recent decades have seen a very rapid success in developing numerical methods based on explicit control over approximation errors. It may be said that nowadays a new direction is forming in numerical analysis, the main goal of which is to develop methods ofreliable computations. In general, a reliable numerical method must solve two basic problems: (a) generate a sequence of approximations that converges to a solution and (b) verify the accuracy of these approximations. A computer code for such a method must consist of two respective blocks: solver and checker.

In this book, we are chiefly concerned with the problem (b) and try to present the main approaches developed for a posteriori error estimation in various problems.

The authors try to retain a rigorous mathematical style, however, proofs are constructive whenever possible and additional mathematical knowledge is presented when necessary. The book contains a number of new mathematical results and lists a posteriori error estimation methods that have been developed in the very recent time.

· computable bounds of approximation errors
· checking algorithms
· iteration processes
· finite element methods
· elliptic type problems
· nonlinear variational problems
· variational inequalities
 

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Contents

Chapter 1 Introduction
1
Chapter 2 Mathematical background
7
Chapter 3 A posteriori estimates for iteration methods
23
Chapter 4 A posteriori estimates for finite element approximations
39
Chapter 5 Foundations of duality theory
79
Chapter 6 Twosided a posteriori estimates for linear elliptic problems
125
Chapter 7 A posteriori estimates for nonlinear variational problems
209
Chapter 8 A posteriori estimates for variational inequalities
245
Bibliography
281
Notation
301
Index
303
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